Partial-dual Euler-genus polynomials for two classes of bouquets

Abstract

[European J. Combin., 2020, 86: Paper No. 103084, 20 pp.] introduced the concept of partial-dual Euler-genus polynomial in the ribbon graphs and gave the interpolation conjecture. That is, the partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating. In fact, [European J. Combin., 2022, 102: Paper No. 103493, 7 pp.] gave two classes of counterexamples to deny the conjecture, and only one or two of the side loops contained in the two classes of bouquets were non-orientable. On the basis of [European J. Combin., 2022, 102: Paper No. 103493, 7 pp.], we further calculate the partial-dual Euler-genus polynomials of two other classes of bouquets. One is non-interpolating, whose side loop has an arbitrary number of non-orientable loops. The other is interpolating, whose side loop has an arbitrary number of both non-orientable loops and orientable loops.

Graphical abstract

Keywords

Ribbon graph / partial-dual / genus / polynomial / interpolating

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